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by Dr. Michael P. Frank, University of Florida Modified by Longin Jan Latecki, Temple University

by Dr. Michael P. Frank, University of Florida Modified by Longin Jan Latecki, Temple University. §12.2 – Finite State Machines with Output. Remember the general picture of a computer as being a transition function T : S × I →S × O ?

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by Dr. Michael P. Frank, University of Florida Modified by Longin Jan Latecki, Temple University

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  1. by Dr. Michael P. Frank, University of Florida Modified by Longin Jan Latecki, Temple University §12.2 – Finite State Machines with Output

  2. Remember the general picture of a computer as being a transition function T:S×I→S×O? • If the state set S is finite (not infinite), we call this system a finite state machine. • If the domain S×I is reasonably small, then we can specify T explicitly by writing out its complete graph. • However, this is practical only for machines that have a very small information capacity.

  3. Size of FSMs • The information capacity of an FSM is C = log |S|. • Thus, if we represent a machine having an information capacity of C bits as an FSM, then its state transition graph will have |S| = 2C nodes. • E.g. suppose your desktop computer has a 512MB memory, and 60GB hard drive. • Its information capacity, including the hard drive and memory (and ignoring the CPU’s internal state), is then roughly ~512×223 + 60×233 = 519,691,042,816 b. • How many states would be needed to write out the machine’s entire transition function graph? 2519,691,042,816 = A number having >1.7 trillion decimal digits!

  4. One Problem with FSMs as Models • The FSM diagram of a reasonably-sized computer is more than astronomically huge. • Yet, we are able to design and build these computers using only a modest amount of industrial resources. • Why is this possible? • Answer: A real computer has regularities in its transition function that are not captured if we just write out its FSM transition function explicitly. • I.e., a transition function can have a small, simple, regular description, even if its domain is enormous.

  5. Other Problems with FSM Model • It ignores many important physical realities: • How is the transition function’s structure to be encoded in physical hardware? • How much hardware complexity is required to do this? • How close in physical space is one bit’s worth of the machine’s information capacity to another? • How long does it take to communicate information from one part of the machine to another? • How much energy gets dissipated to heat when the machine updates its state? • How fast can the heat be removed, and how much does this impact the machine’s performance?

  6. Vending Machine Example • Suppose a certain vending machine accepts nickels, dimes, and quarters. • If >30¢ is deposited, change isimmediately returned. • If the “coke” button is pressed,the machine drops a coke. • It can then accept a new payment. Ignore any otherbuttons, bills,out of change,etc.

  7. Modeling the Machine • Input symbol set: I = {nickel, dime, quarter, button} • We could add “nothing” or  as an additional input symbol if we want. • Representing “no input at a given time.” • Output symbol set:O = {, 5¢, 10¢, 15¢, 20¢, 25¢, coke}. • State set:S = {0, 5, 10, 15, 20, 25, 30}. • Representing how much money has been taken.

  8. Transition Function Table

  9. Transition Function Table cont.

  10. Another Format: State Table Each pair showsnew state,output symbol

  11. Directed-Graph State Diagram • As you can see, these can get kind of busy. q,5¢ d,5¢ q q q,20¢ d d d n n n n n n 0 5 10 15 20 25 30 n,5¢ b b b b b b d,10¢ q,25¢ q,15¢ b,coke q,10¢

  12. Formalizing FSMs • Just like the general transition-function definition from earlier, but with the output function separated from the transition function, and with the various sets added in, along with an initial state. • A finite-state machineM=(S, I, O, f, g, s0) • S is the state set. • I is the alphabet (vocabulary) of input symbols • Ois the alphabet (vocabulary) of output symbols • f is the state transition function • g is the output function • s0 is the initial state. • Our transition function from before is T = (f,g).

  13. Construct a state table for the finite-state machine in Fig. 3. Find the output string for the input 101011

  14. A unit-delay machine: Construct a finite-state machine that delays an input string by one unit of time: Input: x1x2 … xk-1xk Output: 0x1x2 … xk-1 Language recognizer: Construct a finite-state machine that outputs 1 iff the input string read so far ends with 111.

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